While the LSM argument does not apply in dimensions \(d>1\) , it was extended by Oshikawa [1]} and subsequently rigorized by Hastings [2]} to \(d\) -dimensional systems with periodic boundary conditions (i.e. on a \(d\) -torus) and a finite excitation gap. The line of reasoning, inspired by Laughlin's argument for quantization of the Hall conductivity in two dimensions [3]}, focuses on the consequences of adiabatic \(\phi =2\pi \)  \(\textsf {U}(1)\)  flux threading through one of the toroidal cycles, followed by a “pullback" to the original \(\textsf {U}(1)\)  flux state. In the thermodynamic limit, starting with an initial ground state \({| \, \Psi ^{\vphantom{\dagger }}_0(\phi =0) \, \rangle }\) , this procedure results in a low energy state \({| \, \Psi ^{\vphantom{\dagger }}_1(\phi =0) \, \rangle }\)  (similar to the LSM construction), which becomes degenerate with the ground state in the thermodynamic limit. The analysis can be applied to bosonic systems as well (see, e.g., Ref. [4]}, [5]}).
